Survival probability of stochastic processes beyond persistence exponents

For many stochastic processes, the probability S(t) of not-having reached a target in unbounded space up to time t follows a slow algebraic decay at long times, S(t) similar to S-0/t(theta). This is typically the case of symmetric compact (i.e. recurrent) random walks. While the persistence exponent theta has been studied at length, the prefactor S-0, which is quantitatively essential, remains poorly characterized, especially for non-Markovian processes. Here we derive explicit expressions for S-0 for a compact random walk in unbounded space by establishing an analytic relation with the mean first-passage time of the same random walk in a large confining volume. Our analytical results for S-0 are in good agreement with numerical simulations, even for strongly correlated processes such as Fractional Brownian Motion, and thus provide a refined understanding of the statistics of longest first-passage events in unbounded space.